Recurrence equations and their classical orthogonal polynomial ...

314 W. Koepf, D. Schmersau / Appl. Math. Comput. 128 (2002) 303–327
with two possible solutions e ¼ 2a that give the differential equations
and
ðx 2
ðx 2
4Þy 00 ðxÞþ2ðx þ 1Þy 0 ðxÞ nðn 3ÞyðxÞ ¼0; ð27Þ
4Þy 00 ðxÞþ2ðx 1Þy 0 ðxÞ nðn 3ÞyðxÞ ¼0: ð28Þ
They correspond to the densities
rffiffiffiffiffiffiffiffiffiffiffi
4 þ x
qðxÞ ¼
4 x
and
rffiffiffiffiffiffiffiffiffiffiffi
4 x
qðxÞ ¼ ;
4 þ x
respectively, hence the orthogonal polynomials are multiples of the Jacobi
ð1=2; 1=2Þ
ð 1=2;1=2Þ
polynomials Pn ðx=2Þ and Pn ðx=2Þ.
Finally, in the third of the above cases, i.e., for d ¼ 3a, we get again e ¼ 0
and
ðx 2
4Þy 00 ðxÞþ3xy 0 ðxÞ nðn 4ÞyðxÞ ¼0 ð29Þ
corresponding to the density
qðxÞ ¼
1
rðxÞ exp
Z
sðxÞ
rðxÞ
ffiffiffiffiffiffiffiffiffiffiffiffi
dx ¼ 4 x2 p
:
The corresponding orthogonal polynomials are multiples of translated
Chebyshev polynomials of the second kind
pnðxÞ ¼knSnðxÞ ¼ p0
n þ 1 SnðxÞ ¼ p0
n þ 1 Unðx=2Þ ðn P 0Þ ð30Þ
(see e.g. [1], Table 22.2, and (22.5.13); SnðxÞ are monic, see also Table 22.8),
hence
PnðxÞ ¼pn 1ðxÞ ¼ P1
n Un 1ðx=2Þ ðn P 1Þ:
We see that the recurrence equation (24) has four different (shifted) linearly
transformed classical orthogonal polynomial solutions!
Using our implementation, these results are obtained by
> strict: ¼ true:
> RE :¼ ðn þ 3Þ pðn þ 2Þ x ðn þ 2Þ pðn þ 1Þþðn þ 1Þ pðnÞ ¼0;
RE :¼ ðn þ 3Þpðn þ 2Þ xðn þ 2Þpðn þ 1Þþðn þ 1ÞpðnÞ ¼0
> REtoDE(RE,p(n),x);